Problem: $P(x)$ is a polynomial. $P(x)$ divided by $(x-8)$ has a remainder of $5$. $P(x)$ divided by $(x-5)$ has a remainder of $3$. $P(x)$ divided by $(x+5)$ has a remainder of $-2$. $P(x)$ divided by $(x+8)$ has a remainder of $0$. Find the following values of $P(x)$. $P(5)=$
Explanation: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, $P({5})$ is equal to the remainder when $P(x)$ is divided by $(x-{5})$, and we are given that this remainder is equal to $3$. In a similar manner, $P({-8})$ is equal to the remainder when $P(x)$ is divided by $(x-({-8}))$, which can be rewritten as $(x+8)$, and we are given that this remainder is equal to $0$. In conclusion, $P(5)=3$ $P(-8)=0$